*Given a post-NMO constant-offset section at half-offset h_{1}*

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Equation () belongs to the hyperbolic type, with
the offset coordinate *h* being a ``time-like'' variable and the
midpoint coordinate *y* and the time *t*_{n} being ``space-like''
variables. The last condition () is required for the
initial value problem to be well-posed Courant (1962). From a physical
point of view, its role is to separate the two different wave-like
processes embedded in equation (), which are
analogous to inward and outward wave propagation. We will associate
the first process with continuation to a larger offset and the second
one with continuation to a smaller offset. Though the offset
derivatives of data are not measured in practice, they can be
estimated from the data at neighboring offsets by a finite-difference
approximation. Selecting a propagation branch explicitly, for example
by considering the high-frequency asymptotics of the continuation
operators, can allow us to eliminate the need for
condition (). In this section, I discuss the exact
integral solution of the OC equation and analyze its asymptotics.

The integral solution of problem (-) for equation () is obtained in Appendix with the help of the classic methods of mathematical physics. It takes the explicit form

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From equations () and () one can see that the impulse response of the offset continuation operator is discontinuous in the time-offset-midpoint space on a surface defined by the equality

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Figure 7

As a second-order differential equation of the hyperbolic type,
equation () describes two different processes. The
first process is ``forward'' continuation from smaller to larger
offsets, the second one is ``reverse'' continuation in the opposite
direction. These two processes are clearly separated in the
high-frequency asymptotics of operator (). To obtain
the asymptotical representation, it is sufficient to note that is the impulse response
of the causal half-order integration operator and that is asymptotically equivalent to (*t*, *a* >0). Thus, the asymptotical form of
the integral offset-continuation operator becomes

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In the high-frequency asymptotics, it is possible to replace the two terms in equation () with a single term Fomel (1996b). The single-term expression is

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The limit of expression () for the output offset *h*
approaching zero can be evaluated by L'Hospitale's rule. As one would
expect, it coincides with the well-known expression for the summation
path of the integral DMO operator
Deregowski and Rocca (1981)

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12/28/2000